Abstract
Elastic interfaces embedded in (quenched) random media exhibit meta-stability
and stick-slip dynamics. These non-trivial dynamical features have been shown
to be associated with cusp singularities of the coarse-grained disorder
correlator. Here we show that annealed systems with many absorbing states and a
conservation law but no quenched disorder exhibit identical cusps. On the other
hand, similar non-conserved systems in the directed percolation class, are also
shown to exhibit cusps, but of a different type. These results are obtained
both by a recent method to explicitly measure disorder correlators and by
defining an alternative new protocol, inspired by self-organized criticality,
which opens the door to easily accessible experimental realizations.